cp's OEIS Frontend

A multiset partition of weight n is a finite multiset of finite nonempty multisets whose sizes sum to n.

Number of distinct nonnegative integer matrices with all row sums equal and total sum n up to row and column permutations. - Andrew Howroyd, Sep 05 2018

From Gus Wiseman, Oct 11 2018: (Start)

Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears the same number of times. For n = 4, non-isomorphic representatives of these 17 multiset partitions are:

{{1,1,1,1}}

{{1,1,2,2}}

{{1,2,3,4}}

{{1},{1,1,1}}

{{1},{1,2,2}}

{{1},{2,3,4}}

{{1,1},{1,1}}

{{1,1},{2,2}}

{{1,2},{1,2}}

{{1,2},{3,4}}

{{1},{1},{1,1}}

{{1},{1},{2,2}}

{{1},{2},{1,2}}

{{1},{2},{3,4}}

{{1},{1},{1},{1}}

{{1},{1},{2},{2}}

{{1},{2},{3},{4}}

(End)

A non-normal semi-magic square is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

Squares must be of size k X k where k is a divisor of n. This implies that a(p) = p! + 1 for p prime since the only allowable squares are of sizes 1 X 1 and p X p. The 1 X 1 square is [p], the p X p squares are necessarily permutation matrices and there are p! permutation matrices of size p X p. Also, a(n) >= n! + 1 for n > 1. - Chai Wah Wu, Jan 13 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is uniform if all edges have the same size, and regular if all vertices have the same degree. The span of a hypergraph is the union of its edges.

Also the number of 0-1 matrices with n columns, all distinct rows, no zero columns, equal row-sums, and equal column-sums, up to a permutation of the rows.

A non-normal semi-magic rectangle is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

Rectangles must be of size k X m where k and m are divisors of n and k*m >= n. This implies that a(p) = p! + 2 for p prime since the only allowable rectangles are of sizes 1 X 1, 1 X p, p X 1 and p X p. There are no 1 X 1 rectangle that satisfies the condition. The 1 X p and p X 1 rectangles are [1....1] and its transpose, the p X p rectangle are necessarily permutation matrices and there are p! permutation matrices of size p X p. It also shows that a(n) >= n! + 2 for n > 1. - Chai Wah Wu, Jan 13 2019

A coupled non-normal semi-magic rectangle is a nonnegative integer matrix with equal row sums and equal column sums. The common row sum may be different from the common column sum.

Rectangles must be of size k X m where k and m are divisors of n. This implies that a(p) = p! + 3 for p prime since the only allowable rectangles are of sizes 1 X 1, 1 X p, p X 1 and p X p. The 1 X 1 square is [p], the 1 X p and p X 1 rectangles are [1,...,1] and its transpose and the p X p squares are necessarily permutation matrices and there are p! permutation matrices of size p X p. Also, a(n) >= n! + 3 for n > 1. - Chai Wah Wu, Jan 15 2019

A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d, for some d|n.